Compa abili y o U ban S ee Ne wo ks: Conside a ion o he Size E ec in he E alua ion
o Ne wo k Cha ac e is ics and a P oposal o De e mining an App op ia e Size o he
Ca chmen A ea o Pedes ian Ne wo ks
Hsiao-Hui Chen*, Udo Die ich
Resou ce E iciency in A chi ec u e and Planning, Ha enCi y Uni e si y, Hambu g 20457, Ge many
Co esponding Au ho Email: hsiao-hui.chen@hcu-hambu g.de
h ps://doi.o g/10.18280/ijsdp.160603
ABSTRACT
Recei ed: 23 May 2021
Accep ed: 19 Oc obe 2021
Fo he analysis o u ban ne wo ks indica o s, like a e age node deg ee, connec i i y and
be weenness cen ali y, a e widely used. Thei alues a e calcula ed o a ca chmen a ea o
he ne wo k. This esea ch in es iga es i and o wha ex en he size o he ca chmen a ea
in luences he indica o ’s alues. A me hodology o de e mining he size o ca chmen a ea is
p oposed and ecommenda ions o i s app op ia e size o pedes ian ne wo k analysis a e
o e ed. Fo he ma hema ical deduc ion o he ela ionship be ween he size o he ca chmen
a ea and he indica o alues an idealized egula ne wo k is used as a e e ence model. We
ound ha he size e ec on indica o alues is p ominen . I dec eases exponen ially as he
size o he ca chmen a ea inc eases. The size e ec is no able un il a side leng h o he
ca chmen a ea ha equals 50 imes he a e age s ee leng h, while he a ia ion is s ill
accep able i i is 30 o 40 imes he a e age s ee leng h. The ypical a e age s ee leng h in
ci ies is abou 100 me e s, a ca chmen a ea la ge han 4000 x 4000m2 is no necessa y. This
e e ence model can also be used o e alua e eal ne wo ks and o compa e hem wi h an
idealized case.
Keywo ds:
a e age s ee leng h, ca chmen a ea,
edge e ec , ne wo k analysis, node
deg ee, size e ec
1. INTRODUCTION
1.1 The challenge
When e alua ing he eal-wo ld spa ial ne wo ks, he cen al
poin and he ca chmen a ea a ound i need o be de ined
be o e ca ying ou ne wo k analysis, so ha he indica o s o
di e en ne wo ks can be compa ed wi h each o he . Howe e ,
he eal-wo ld spa ial ne wo ks a e no s ic ly disc ee sys ems
and he bounda ies o hei ca chmen a eas a e o en
a bi a ily de ined. This a bi a ily de ined size o he
ca chmen a ea o en exe s signi ican dis o ion on he alues
o ne wo k indica o s [1-6] o he ollowing easons.
I he size o he ca chmen a ea is oo la ge, i is mo e likely
ha i con ains mul iple sub-ne wo ks o sub-s uc u es wi h
di e en cha ac e is ics. In o he wo ds, he mixed
cha ac e is ics o he en i e ne wo k a e o en a combina ion o
sub-s uc u es, such as g ids, ees, hubs and spokes, o lines
s uc u e. And each ype o ne wo k s uc u e may di e in
complexi y [7, 8]. Since he alue o he indica o is he
a e age alue o he en i e ne wo k, he complexi ies o he
sub-s uc u es wi hin he ne wo k may a ec he calcula ion o
he indica o o he e alua ion o he cha ac e is ics and
a ibu es.
On he o he hand, i he ca chmen a ea is oo small, he
ne wo k model excludes he cha ac e is ics beyond he
a bi a ily de ined bounda y o he model. Since he analy ic
algo i hms a e ela ional, he wo ld ou side he ca chmen a ea
o analysis s ill a ec s he wo ld inside i and he mo emen
pa e ns wi hin [1].
Conce ns o e he size e ec on he eliabili y o
signi icance o he ne wo k analysis esul s ha e been
exp essed and sha ed by many esea che s [1, 9-12]. They es
he size e ec on he pe o mance o spa ial ne wo k models
by a ying he h eshold adius o he ca chmen a ea. Fo
example, Yoshimu a e al. [13] ha e in es iga ed be weenness
cen ali y wi h adius o he ca chmen a ea a ying om 300
o 5000 me e s wi h 100 me e s s ep, and he esul s show ha
he indica o alue is sensi i e o he size o he ca chmen a ea.
To be mo e speci ic, he indica o alue o he same node o
link in he ne wo k may change wi h he size o he ca chmen
a ea. And his dis o ion is pa icula ly p onounced o he
nodes and links a he bo de o he ca chmen a ea [1, 14]. Fo
example, Gil [1] has ound ha nodes o links nea he cen e
o he ca chmen a ea end o ha e highe deg ee o
be weenness cen ali y compa ed wi h hose close o he
bo de . Acco dingly, his in luence has been called he “edge
e ec ” o “bounda y e ec ” [1, 9, 10, 14].
In his esea ch, we ocus on he e ec o size o he
ca chmen a ea on he indica o alue. This e ec is he ea e
e e ed o as he “size e ec ”.
1.2 Types o esea ch whe e size e ec should be
conside ed
Conside a ion o he size e ec may be pa icula ly c ucial
in esea ch p ojec s associa ed wi h he ollowing pu poses.
1.2.1 Resea ch aiming a compa ing and classi ying mul iple
ne wo ks dis ibu ed in di e en loca ions
Fo any compa a i e s udy, whe e mo e han one ne wo k is
In e na ional Jou nal o Sus ainable De elopmen and
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1019
included, he size o he ca chmen a ea will ei he ha e o be
iden ical o i mus be examined as one o he ac o s ha may
explain he a ia ion o he measu emen alues among he
mul iple ne wo ks.
1.2.2 Resea ch equi ing he dis inc ion among di e en ypes
o mobili y ne wo ks based on he mode o anspo a ion,
such as he ne wo ks o pedes ian, cyclis s o mo o ized
ehicles
Fo example, “a small oad segmen in a esiden ial a ea
migh be impo an o pedes ians, bu i is almos negligible
o mo o ized anspo a ion. In his sense, he oad segmen
should ha e a leas wo kinds o impo ance: one o
pedes ians and he o he o mo o ized anspo a ion” [15].
In o he wo ds, he alue o he indica o a ies depending on
whe he he ca chmen a ea is a he human o ehicle scale.
Wi hou conside ing he size o he s udy a ea, he deg ee o
be weenness cen ali y canno ep esen mul iple aspec s o he
ci y ha is pe cei ed by people in he eal wo ld [16]. To a oid
hese p oblems, Yoshimu a e al. [13] p opose ha , in addi ion
o he global be weenness cen ali y o he en i e ci y, a se o
local be weenness cen ali y alues a a smalle neighbo hood
scale shall also be calcula ed.
1.2.3 Resea ch whe e an a ea needs o be di ided in o sec ions
o smalle size wi h egula shape
The ci y a ea is o en di ided in o smalle uni s due o he
s uc u al complexi y and size associa ed wi h he scale
p oblem [17-19]. Fo example, he analysis o connec i i y o
he oad and s ee ne wo k s uc u e is o en conduc ed wi h
he applica ion o opological measu es and ypically equi es
ha a la ge geog aphical space be di ided in o smalle pa s
wi h egula shape [8]. In a ecen s udy, in o de o de e mine
he size o he ca chmen a ea on he alue o he selec ed
measu es, Soczówka e al. [8] ha e ca ied ou a compa a i e
analysis by di iding he analyzed a ea in o di e en sizes wi h
egula shape and compa ing he alues o connec i i y
measu emen s o he oad and s ee ne wo k s uc u e. In such
a case, he size o a single basic ca chmen a ea is e y
impo an because i can signi ican ly a ec he compu a ional
esul s o hese measu es.
1.2.4 Resea ch measu ing he spa ial accessibili y
A bi a y adminis a i e bounda ies (such as census ac s
o block g oups) a e o en used in he s udies o spa ial
accessibili y. The a bi a ily de ined bo de may lead o a
me hodological limi a ion [20, 21] because, i s o all,
accessibili y in ol es mo emen and a gi en bounda y does
no ac ually p e en people o ehicles a eling ac oss he
bo de om eaching he acili ies, se ices, ameni ies o any
poin s o in e es [22]. Secondly, a gi en bounda y may
exclude beha io ou side he ca chmen a ea [12, 20-22, 23-
31]. Thi dly, he esou ce o se ices beyond a de ined
a bi a y bounda y may in luence he beha io wi hin he
ca chmen a ea [27]. Many esea ch p ojec s s udying he
spa ial accessibili y ha e poin ed ou he isk ha he
accessibili y o acili ies may be biased [22, 25, 27, 32] o
unde - epo ed [12, 23, 28, 29, 33] because “a eas close o he
bounda y may be classi ied as ha ing poo geog aphic access
e en hough hey may in ac be p oxima e o esou ces ac oss
he bounda y” [34].
1.3 P oposed solu ions o mi iga ing he dis o ion
In an a emp o mi iga e, con as o diminish he dis o ions
s emming om he size e ec , he p oposed ea u es and
p inciples unde lying u he classi ica ion can be b oadly
ca ego ized in o he ollowing wo app oaches.
1.3.1 Adding a bu e zone wi h he assigned h eshold adius
The i s app oach is o c ea e bu e zones a ound he
ca chmen a ea in o de o compu e he indica o alue [1, 4,
35-37]. Fo example, he Py hon lib a y OSMnx [38, 39]
au oma ically c ea es a bu e o hal a kilome e a ound he
eques ed a ea so ha each node has a co ec s ee coun . The
bu e zones a e hen immed om he cons uc ed ne wo k
model. The indica o alues o nodes and links wi hin he
bu e zone a e excluded om he analysis because hey a e
no eliable.
1.3.2 Using a homogeneous ea u e as he c i e ion o he
di ision
Soczówka e al. [8] p opose o di ide he ci y a ea in o
smalle ca chmen a eas wi h homogeneous ea u es be o e
conduc ing he analysis. They sugges se e al possible c i e ia
o he homogeneous ea u es including, i s ly, adminis a i e
c i e ia, such as adminis a i e dis ic bounda ies; secondly,
s uc u al c i e ia, such as he spa ial dis ibu ion o densi y o
inhabi an s in households; and, hi dly, echnical and
unc ional c i e ia, such as he public anspo managemen
sys em.
In sho , any di isions and classi ica ions o geog aphical
space should de ine he bounda y o he ca chmen a ea. In
esponse o hese conside a ions, he cu en esea ch in ends
o con ibu e o his decision p ocess and o quan i y he e ec
on he alue o he indica o s by 1) in es iga ing whe he and
o wha ex end indica o s, such as he node deg ee, change
wi h he size o he ca chmen a ea; 2) p oposing a
me hodology o decide he size o he ca chmen a ea based on
he a e age s ee leng h; and 3) o e ing he ecommenda ion
o he app op ia e size o he ca chmen a ea o he
in es iga ion o pedes ian ne wo ks.
A heo e ical model ep esen ing an idealized egula
ne wo k (see Figu e 1) has been c ea ed as a e e ence model
o he clea ma hema ical deduc ion o he ela ionship
be ween he size o he ca chmen a ea and he changes in he
indica o alues. This allows us o answe he ollowing
esea ch ques ions in a mo e con olled en i onmen , whe e
he di e ences in he alues o he indica o will only be
caused by he size e ec (This is no igno ing he ac ha , in
he case o eal ne wo k, he size e ec may s ill exis
ega dless o he size o he ca chmen a ea. Because a eal
ne wo k is no egula o e e ni y and, he e o e, i s
cha ac e is ics change).
• How do we decide he app op ia e size o he
in es iga ion o pedes ian ne wo ks?
• A e he e uppe and lowe limi s o he size e ec on
he alue o an indica o ?
• How big should he ne wo k ca chmen a ea be in
o de o be able o compa e i wi h he e e ence
model?
1020
2. METHOD
2.1 The in es iga ed indica o
The indica o we ha e chosen o examine he size e ec is
he a e age node deg ee, k, which is exp essed by
𝑘 = 2 × 𝑀
𝑁
(1)
whe e, M e e s o he o al numbe o links and N e e s o he
o al numbe o nodes. In addi ion, he a e age node deg ee
p o ides he in o ma ion o he ne wo k pa e n and can also
be used o e alua e he le el o "g idness.". A ne wo k wi h a
lo o nodes connec ing o 4 links, i.e. k  4, means ha he
ne wo k is mo e likely o be g id-pa e n. And "mo e-g idded
ci ies ha e highe connec i i y (i.e., highe node deg ees, mo e
ou -way in e sec ions, ewe dead-ends e c.) and less-winding
s ee pa e ns"[39]. This means ha , holding he numbe o
nodes cons an , i he a e age node deg ee o a eal ne wo k is
smalle han ha o a g id-pa e n ne wo k, he connec i i y o
he eal ne wo k is wo se han he connec i i y o a g id-
pa e n ne wo k.
Be o e examining he size e ec on he node deg ee, we
need o p o ide some de ini ions conce ning ou heo e ical
model.
2.2 The heo e ical ma hema ical model
Fo he in es iga ion o he size e ec , a eal ne wo k migh
no be a good basis o be he e e ence o compa ison. The
main p oblem is ha he size e ec canno be clea ly sepa a ed
om o he e ec s. The e o e, a heo e ical model o he
p ognosis o he size e ec was c ea ed in o de o p o ide he
me hod o ma hema ical analysis o an idealized egula
ne wo k. In his way, he di e ences in he alues o he
indica o will only be caused by he size e ec . Ou de ini ion
o an idealized ne wo k consis s o he ollowing componen s.
2.2.1 De ini ion o a single quad a as he basic elemen
To illus a e he ules o how he ne wo k is c ea ed, we s a
wi h a ne wo k ha is jus a quad a (squa e). As shown in (a)
in Figu e 2 and Table 1, a quad a has ou links as i s bounda y
and i se e as he basic uni o he quad a ic ne wo k wi h ou
nodes (N = 4) and ou links (M = 4). We use d o e e o he
leng h o a link in he ne wo k, which is also he side leng h o
a quad a .
2.2.2 De ini ion o a ne wo k
A quad a is expanded and ex ended in he same quad a ic
pa e n, as shown in (b), (c) and (d) in Figu e 2 and Table 1.
The size o he ne wo k is measu ed by he o al numbe o
nodes, N, and he o al numbe o links, M. Elemen (b) in
Figu e 2 and Table 1 show ha a ne wo k consis ing o ou
quad a s has nine nodes (N = 9) and 12 links (M = 12).
2.2.3 De ini ion o a ca chmen a ea
In he case o eal ne wo ks, a ne wo k is di e en om a
ca chmen a ea and he selec ed ca chmen a ea will be smalle
han he en i e ne wo k. As illus a ed in Figu e 1 (a) and (b),
he black solid line indica es he links and also he bounda y
o a quad a ic ne wo k and he ed dashed line indica es he
bounda y o a ca chmen a ea. A ca chmen a ea is like he
sec ion ha cu s o links connec ing nodes om inside o he
ca chmen a ea o nodes ou side o he ca chmen a ea (In a
eal ne wo k, i is no necessa ily he case ha a ca chmen a ea
cu s o links connec ing nodes om inside o he ca chmen
a ea o nodes ou side o he ca chmen a ea. Some imes, he
bounda y o he ca chmen a ea may coinciden ly lie exac ly
on a link as shown in Figu e 1(b).). The link cu o by he
ca chmen a ea could be coun ed as a comple e o a hal link.
And he alue o he indica o will be a ec ed by how hese
links a e coun ed. Al e na i ely, hey migh be excluded om
he calcula ion al oge he . Fo he pu pose o his pape , we
assume ha hese cu -o links a e neglec ed and hey a e
excluded om he calcula ion.
This means ha he bounda y o he ca chmen a ea consis s
o he links o he ne wo k as shown in Figu e 1(b). The e o e,
he side leng h o a ca chmen a ea, D, consis s o one o
mul iple links o he ne wo k. And he size o he ca chmen
a ea is D x D.
(a)
(b)
Figu e 1. Abs ac ed exp ession o he bounda ies o he
idealized ne wo k and he ca chmen a ea
The e can be wo kinds o ela ionships be ween he
ne wo k and he ca chmen a ea.
2.2.4 When he ca chmen a ea consis s o one quad a and D
consis s o single d (i.e. D = d)
In he case o he smalles ca chmen a ea (as shown in
elemen (a) in Figu e 2 and Table 1), he e is only one quad a
and, he e o e, he side leng h, D, o his ca chmen a ea equals
o d. The size o he ca chmen a ea is D×D = d2. The ne wo k
in his ca chmen a ea has 4 links (M = 4) and 4 nodes (N = 4).
The node deg ee, k, is acco dingly 2.
2.2.5 When he ca chmen a ea consis s o mul iple quad a s
and D consis s o mul iple d
In elemen (b) in Figu e 2 and Table 1, he e a e 4 quad a s
and, he e o e, he side leng h, D, o his ca chmen a ea equals
o 2d. The size o he ca chmen a ea is D x D = 4d2. The
ne wo k in his ca chmen a ea has 12 links. The o al numbe
Is his a
comple e o
hal link, o
no ega ded
as a link a
all?
An idealized
ne wo k wi h
links (black
solid line) as i s
bounda y.
A ca chmen
a ea wi h ed
dashed line
indica ing i s
bounda y.
1021
o links, M, in his ne wo k is 12 and he o al numbe o nodes,
N, is 9. The node deg ee, k, is acco dingly 2.67.
The ela ionship be ween he quad a , he ne wo k and he
ca chmen a ea o a ious sizes a e summa ized in Table 1 and
i shows ha , wi h he inc easing side leng h, D, o he
ca chmen a ea, he node deg ee, k, also inc eases. This means
ha he e is a size e ec on he indica o o node deg ee. This
end is no ewo hy, so we b eak down he p ocess in o s eps
in o de o in es iga e he de ails. The ollowing discussion is
di ided in o wo pa s: he addi ional links and he addi ional
nodes.
Figu e 2. Abs ac ed exp ession o he p ocess o inc easing
he size o he ca chmen a ea
Table 1. Rela ionship be ween he quad a , ne wo k and
ca chmen a ea o a ious sizes
Elemen s in Figu e 2
(a)
(b)
(c)
(d)
Numbe o quad a s
1
4
16
36
Ne wo k
To al
numbe o
links (M)
4
12
40
84
To al
numbe o
nodes (N)
4
9
25
49
Node
deg ee (k)
2
2.67
3.2
3.43
Ca chmen
a ea
Side leng h
o
ca chmen
a ea (D)
d
2d
4d
6d
Size o
ca chmen
a ea (D×D)
d2
4d2
16d2
36d2
2.3 P ocess o inc easing he size o he quad a ic ne wo k
2.3.1 Inc easing he numbe o links
Figu e 2 p esen s he abs ac ed exp ession o he p ocess o
inc easing he size o he ca chmen a ea. Figu e 2 (a) shows
one quad a (squa e). The ne wo k wi h one quad a is he
smalles ne wo k.
Figu e 2 (b) shows a ne wo k consis s o 4 quad a s. To
ex end he ne wo k om (a) o (b) in Figu e 2 and Table 1,
h ee links a e added o c ea e he yellow quad a s. And hen
wo mo e links a e added o c ea e he blue quad a .
Fo he ne wo k in (c) in Figu e 2 and Table 1, which is a
ne wo k consis s o 16 quad a s, we begin wi h a co ne and,
i s ly, c ea e he g een quad a e wi h ou links. Nex , wo
mo e links a e added o c ea e ano he wo blue quad a s.
Thi dly, h ee links a e added o c ea e he yellow quad a in
he co ne . Finally, hese h ee s eps a e epea ed un il he inal
pu ple quad a e, which needs only one addi ional link o be
c ea ed.
As he ne wo k becomes bigge and bigge , he e a e mo e
and mo e blue quad a s, which a e o med by wo addi ional
links, among he newly c ea ed quad a s. In o he wo d, he
numbe o blue quad a s inc eases much as e han o he
quad a s. E en ually his ype o quad a becomes mo e and
mo e impo an and hus dominan he pa e n o he inc easing
side leng h, D. Meanwhile, he yellow co ne quad a , which
consis s o 3 links, becomes less dominan .
2.3.2 Inc easing he numbe o nodes
Since he blue quad a is mo e dominan han he o he ypes
o quad a s on he inc easing side leng h, D, he ocus o he
in es iga ion is on his ype o quad a s. Fo e e y blue quad a ,
i akes wo addi ional links o c ease one addi ional node.
E en ually he o al numbe o nodes, N, will be hal o he
o al numbe o links, M. The e o e,
𝑁 = 1
2𝑀
(2)
o , in o he wo ds,
𝑀 = 2 𝑁
(3)
This means ha node deg ee, k, will e en ually app oach he
inal limi .
𝑘 = 2 × 𝑀
𝑁= 2 × 2𝑁
𝑁= 4
(4)
The esul s a e shown in Figu e 3 and Table 2.
Figu e 3. Rela ionship be ween he alue o he node deg ee,
k, and he size o he ca chmen a ea, which is exp essed by
he numbe o links on each side o he ca chmen a ea
2.4 App op ia e size o he ca chmen a ea o pedes ians
Fo he in es iga ions o pedes ian ne wo ks, i makes no
sense o explo e a la ge a ea. I we conside ha he
pedes ians’ maximum accep able walking ime is abou 15 o
20 minu es, he size o he ca chmen a ea would be be ween
1500x1500m² o 2000x2000m². Also, ollowing he indings
ega ding he size e ec , he compa ison be ween di e en
“pedes ian” ne wo ks is only co ec and hus possible i all
ca chmen a eas ha e he same size.
2.00
2.50
3.00
3.50
4.00
010 20 30 40 50 60 70 80 90 100
node deg ee, k
Numbe o links, which is equi alen o D/d, on each
side o he ca chme a ea.
1022
Table 2. Rela ionship be ween size o he ca chmen a ea and he alues o indica o s. The size o he ca chmen a ea is indica ed
by i s side leng h, D, and D is indica ed by he numbe o links, which is equi alen o D/d
Side leng h, D, o he
ca chmen a ea
Links in he ne wo k
Numbe o links on each side
o he ca chmen a ea
To al numbe o links, M
To al addi ional links o
base quad a , indica ed by
he black links
To al addi ional links equi ed o c ea e he
new quad a , indica ed by colo ul links
1
4
4
4
2
12
4
8
4
40
12
28
6
84
40
44
8
144
84
60
10
220
144
76
12
312
220
92
14
420
312
108
16
544
420
124
18
684
544
140
20
840
684
156
40
3280
2964
316
50
5100
4704
396
100
20200
19404
796
Side leng h, D, o he
ca chmen a ea
New quad a
Numbe o links on each side
o he ca chmen a ea
Numbe o new g een
quad a s o med by 4
addi ional links
Numbe o new yellow
quad a s o med by 3
addi ional links
Numbe o new blue
quad a s o med o 2
addi ional links
Numbe o
pu ple quad a s
o med o 1
addi ional link
1
1
0
0
0
2
0
2
1
0
4
1
3
7
1
6
1
3
15
1
8
1
3
23
1
10
1
3
31
1
12
1
3
39
1
14
1
3
47
1
16
1
3
55
1
18
1
3
63
1
20
1
3
71
1
40
1
3
151
1
50
1
3
191
1
100
1
3
391
1
Side leng h, D, o he
ca chmen a ea
Indica o s
Numbe o links on each side
o he ca chmen a ea
To al numbe o nodes, N
Node deg ee, k
Nodes / a ea
A ea
1
4
2.00
4.00
1
2
9
2.67
2.25
4
4
25
3.20
1.56
16
6
49
3.43
1.36
36
8
81
3.56
1.27
64
10
121
3.64
1.21
100
12
169
3.69
1.17
144
14
225
3.73
1.15
196
16
289
3.76
1.13
256
18
361
3.79
1.11
324
20
441
3.81
1.10
400
40
1681
3.90
1.05
1600
50
2601
3.92
1.04
2500
100
10201
3.96
1.02
10000
3. RESULTS AND IMPLICATIONS
The bene i o in es iga ing he size e ec wi h a heo e ical
model is clea because, wi h he con olled condi ion, we can
ind ou he scena io when he size e ec is (nea ly) sa u a ed
and deli e a co ec p ognosis abou he o al size o he
ne wo k. Figu e 3 shows he ela ionship be ween he alue o
he node deg ee, k, and he size o he ca chmen a eas, which
is exp essed by he numbe o links on each side o he
ca chmen a ea. I can be concluded ha he size e ec is e y
no able un il he side leng h o he ca chmen a ea, D, equals
10 imes o he leng h o he link, i.e. D = 10d. And i is s ill
1023

no able wi h a de i a ion o 3.8/4 = 5% when D = 20d. The
de i a ion is educed o 2.5% when D=40d, which can be
ega ded as a h eshold and ecommended as he size ha is
big enough o allow compa isons be ween di e en ca chmen
a eas and/o di e en ne wo ks. E en ually, i comes close o
sa u a ion whe e D = 50d. Fo alues D >= 50d he node
deg ee, k, eaches he ideal alue o he ex ended ne wo k and
he size does no play a ole a e ha .
The p incipal implica ion o his heo e ical model is
wo old. Fi s o all, he abo e in es iga ion p o ides e idence
o show ha he size e ec is ema kable and canno be
neglec ed un il D = 40d.
Secondly, Table 3 shows he ela ionship be ween he
a e age s ee leng h and he ca chmen a ea size ange. The
esul s o e a p inciple guideline o de e mining he size o
he ca chmen a ea whe e he eal ne wo k can be compa ed
wi h he heo e ical model. In he eal s ee ne wo k, he
a e age leng h o a s ee , which is he leng h o he link, d, in
he heo e ical model, is mos ly be ween 50m and 100m.
Assuming ha d = 100m, he size o he selec ed ca chmen
a ea has o be a leas 20d x 20d = 2000x2000m2 in o de o be
able o compa e i wi h he heo e ical model. I he side leng h
o he ca chmen a ea is 40d, i.e. 4000m, and he size o he
ca chmen a ea is a leas 4000x4000m2, he size e ec will be
e en less signi ican in he heo e ical model. The e o e, we
p o ide he e idence o suppo ha he lowe and uppe limi s
o he size o he ca chmen a ea should be 2000x2000m2 and
4000x4000m2.
Table 3. Recommenda ions o he lowe and uppe limi s o
he side leng h o he ca chmen a ea
A e age s ee
leng h, d
50m
100m
Lowe limi o he
side leng h o he
ca chmen a ea
20d=1000m
20d=2000m
Uppe limi o he
side leng h o he
ca chmen a ea
40d=2000m
40d=4000m
Range o he sizes o
ca chmen a eas
1000x1000m2 ~
2000x2000m2
2000 x 2000m2 ~
4000 x 4000m2
To sum up, he esul s in his esea ch con ibu e o he
a gumen ha a ca chmen a ea wi h an a ea size ha is oo
la ge would no be p ac ical due o he ollowing easons.
Fi s o all, he cha ac e is ics and pa e ns o he s ee
ne wo ks in a eal ci y may a y om qua e o qua e . Fo
example, he cha ac e is ics o he ne wo k in he his o ical
cen e would be di e en om i s su ounding a eas.
The e o e, i he size o he ca chmen a ea is oo big, he
alues o he indica o s would e lec no he in o ma ion abou
one ype o ne wo k bu a he abou a sum o mul iple ypes
o ne wo ks in se e al neighbo ing and connec ed qua e s.
Such mixed in o ma ion would be less aluable o
in es iga ing he ela ionship be ween he s ee ne wo k
s uc u e and he indica o s o o classi ying he s ee
ne wo ks.
Secondly, he a e age s ee leng h can be one o he
indica o s o de e mining he size o he ca chmen a ea.
Acco ding o ou analysis, in he heo e ical ne wo k, he size
e ec on he indica o is no e y signi ican when he size o
he ca chmen a ea is la ge han 4000 x 4000m2. The e o e,
any size la ge han 4000 x 4000m2 would no be necessa y.
Thi dly, he node deg ee o an idealized egula ne wo k
changes wi h he size o he ca chmen a ea. Bu he a ia ion
becomes nea ly neglec able when he size o he ca chmen
a ea is la ge han D = 40d and anishes wi h D >= 50d. This
means ha , i he size o he ca chmen a ea o he eal ne wo k
is la ge han D = 40d o 50d and he node deg ee is 4, he
ne wo k pa e n has a g id-like cha ac e . Howe e , i he node
deg ee o a eal ne wo k is smalle han 4, i s connec i i y is
wo se han ha o he heo e ical g id-pa e n ne wo k and ice
e sa.
Fou hly, we sugges ha in all u u e in es iga ions he size
o he ca chmen a ea should be de ined be o e ca ying ou
u he analysis.
Fi hly, when compa ing he indica o s o mul iple
ca chmen a eas, all o he ca chmen a eas should ha e he
same size as long as D is less han 40d o 50d.
Finally, node deg ee has been chosen because i s beha io
can be calcula ed o he heo e ical and idealized ne wo ks
p oposed in he cu en esea ch in o de o quan i y and o
demons a e he size e ec on he indica o alues s ep by s ep.
To wha ex en o he indica o s each he sa u a ion un il he
size e ec anishes should be pa o u he in es iga ions. I
would be in e es ing o es whe he hei h eshold is also
a ound D = 40 o 50 d. In u u e esea ch, he analysis o he
a iabili y and he sensi i i y o indica o s shall acili a e
decisions on which indica o s should be used o a gi en size
o he ca chmen a ea.
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NOMENCLATURE
d
leng h o a link in he ne wo k and also he side
leng h o a quad a , m
D
side leng h o a ca chmen a ea, m
D×D
size o ca chmen a ea
k
dimensionless a e age node deg ee
M
o al numbe o links
N
o al numbe o nodes
1026